Linear solvers for the finite pointset method

Many simulations in Computational Engineering suffer from slow convergence rates of their linear solvers. This is also true for the Finite Pointset Method (FPM), which is a Meshfree Method used in Computational Fluid Dynamics. FPM uses Generalized Finite Difference Methods (GFDM) in order to discretize the arising differential operators. Like other Meshfree Methods, it does not involve a fixed mesh; FPM uses a point cloud instead. We look at the properties of linear systems arising from GFDM on point clouds and their implications on different types of linear solvers, specifically focusing on the differences between one-level solvers and Multigrid Methods, including Algebraic Multigrid (AMG). With the knowledge about the properties of the systems, we develop a new Multigrid Method based on point cloud coarsening. Numerical experiments show that our Multicloud method has the same advantages as other Multigrid Methods; in particular its convergence rate does not deteriorate when refining the point cloud. In future research, we will examine its applicability to a broader range of problems and investigate its advantages in terms of computational performance.